Problem 4 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the value of the indicated sum. $$ \sum_{l=3}^{8}(l+1)^{2} $$

Step-by-Step Solution

Answer

The sum is 271.

Step 1: Understand the Summation Notation

The expression \( \sum_{l=3}^{8}(l+1)^{2} \) tells us to evaluate the sum of \( (l+1)^2 \) as \( l \) ranges from 3 to 8. This means we need to calculate \( (4)^2, (5)^2, (6)^2, (7)^2, (8)^2, \) and \( (9)^2 \), and then sum these results.

Step 2: Calculate Each Term

Evaluate each term:- For \( l = 3 \), \( (l+1)^2 = (3+1)^2 = 4^2 = 16 \)- For \( l = 4 \), \( (l+1)^2 = (4+1)^2 = 5^2 = 25 \)- For \( l = 5 \), \( (l+1)^2 = (5+1)^2 = 6^2 = 36 \)- For \( l = 6 \), \( (l+1)^2 = (6+1)^2 = 7^2 = 49 \)- For \( l = 7 \), \( (l+1)^2 = (7+1)^2 = 8^2 = 64 \)- For \( l = 8 \), \( (l+1)^2 = (8+1)^2 = 9^2 = 81 \)

Step 3: Sum the Calculated Terms

Add the results from each calculation:\( 16 + 25 + 36 + 49 + 64 + 81 \).Evaluate:\( 16 + 25 = 41 \) \( 41 + 36 = 77 \) \( 77 + 49 = 126 \) \( 126 + 64 = 190 \) \( 190 + 81 = 271 \)

Step 4: Write the Final Answer

The sum of the series from \( l=3 \) to \( l=8 \) for \( (l+1)^2 \) is \( 271 \).

Key Concepts

series evaluation

Evaluating a series involves finding the total sum of a sequence of numbers. In our exercise, evaluating the series means summing the results of \( (l+1)^2 \) as the variable \( l \) changes from 3 to 8. Series evaluation is like a math treasure hunt, where each number gives us a hint until we find the treasure, which is the final sum.

• Step 1: Identify the range, here it's from \( l=3 \) to \( l=8 \).
• Step 2: Calculate the expression \( (l+1)^2 \) for each value of \( l \) in the range.
• Step 3: Add all calculated results to find the series sum.

Imagine evaluating a series as climbing a flight of stairs with each step representing a new calculated term. Reaching the top gives you the entire summation value.

summation notation

Summation notation is a compact way to illustrated adding up a series of numbers. It is usually represented by the Greek letter sigma (\( \sum \)). In our exercise, \( \sum_{l=3}^{8}(l+1)^{2} \) tells us to sum certain terms from 3 to 8 within the given expression. Summation notation is a staple in calculus for organizing computations neatly and efficiently.

• Notice the limits of the sum: here, from 3 to 8.
• Understand each term, \( (l+1)^2 \), in the expression.
• Follow the summation direction from lower index to upper index.

By using summation notation, mathematicians can simplify and solve complex problems more easily. It conveys a lot of information in a small space, helping us keep everything in order.

mathematical series

A mathematical series is essentially the sum of the terms of a sequence. It’s like connecting the dots between numbers to complete a picture. In this exercise, we're building a series of square numbers with \( (l+1)^2 \) and adding them up.

• A sequence is a list of numbers generated by a rule.
• A series takes that sequence and finds the total sum of it.
• The terms in our series are \( (4)^2, (5)^2, (6)^2, (7)^2, (8)^2, \) and \( (9)^2 \).

Think of a series as assembling different pieces. Each number is part of a larger story that, when summed, reveals the overall picture or, in our case, the total value of \( 271 \).

algebraic summation

Algebraic summation is the art of adding up numbers using algebra. It's not just about getting numbers together; it's about doing it elegantly using algebraic expressions and techniques. In our exercise, we're using the expression \( (l+1)^2 \) to derive each term algebraically for the given range.

• Recognize the pattern or rule given by \( (l+1)^2 \).
• Calculate each term systematically.
• Add all terms to obtain the final sum.

Algebraic summation is like weaving a tapestry where each thread (term) contributes to the complete pattern (sum). This technique is fundamental in understanding growth patterns, financial calculations, and many other real-world applications.